X-ray Diffraction

Interaction of Waves

Reciprocal Lattice and Diffraction

X-ray Scattering by Atoms

The Integrated Intensity

Basic Principles of Interaction of Waves

Periodic waves characteristic:

Frequency : number of waves (cycles) per unit time – =cycles/time.

[] = 1/sec = Hz.

Period T: time required for one complete cycle – T=1/=time/cycle. [T] = sec.

Amplitude A: maximum value of the wave during cycle.

Wavelength : the length of one complete cycle. [] = m, nm, Å.

tx

Atc

xA

tkxAxtE

2exp2exp

exp),(

2,2 k

A simple wave completes cycle in 360 degrees

Basic Principles of Interaction of Waves

Consider two waves with the same wavelength and amplitude but displaced a distance x0.

The phase shift:

tAtE exp)(1

02

x

x0

tAtE exp)(1 Waves #1 and #2 are 90o out of phase

Waves #1 and #2 are 180o out of phase

When similar waves combine, the outcome can be constructive or destructive interference

Superposition of Waves

Resulting wave is algebraic sum of the amplitudes at each point

Small difference in phase Large difference in phase

Superposition of Waves

Thomas Young's diagram of double slit interference (1803)

𝜃𝑓 ≈𝜆

𝑑The angular spacing of the fringes is given by 𝜃𝑓 ≪ 1, where

d

Δ𝑥 =𝐿𝜆

𝑑

L

The discovery of X-ray diffraction and its use as a probe of the structure of matter

First diffraction pattern from NaCl crystal

Max von Laue

• The reasoning: x-rays have a wavelength similar to the interatomic distances in crystals, and as a result, the crystal should act as a diffraction grating.

• 1911, von Laue suggested to one of his research assistants, Walter Friedrich, and a doctoral student, Paul Knipping, that they try out x-rays on crystals.

• April 1912, von Laue, Friedrich and Knipping had performed their pioneering experiment on copper sulfate.

The discovery of X-ray diffraction and its use as a probe of the structure of matter

• They found that if the interatomic distances in the crystal are known, then the wavelength of the X-rays can be measured, and alternatively, if the wavelength is known, then X-ray diffraction experiments can be used to determine the interplanar spacings of a crystal.

• The three were awarded Nobel Prizes in Physics for their discoveries.

ZnS Laue photographs along four-fold and three-fold axes

Friedrich & Knipping's improved set-up

The Laue Equations

If an X-ray beam impinges on a row of atoms, each atom can serve as a source of scattered X-rays.

The scattered X-rays will reinforce in certain directions to produce zero-, first-, and higher-order diffracted beams.

The Laue Equations

Consider 1D array of scatterers spaced a apart.

Let x-ray be incident with wavelength .

ha 0coscos

lc

kb

ha

0

0

0

coscos

coscos

coscos

2D

3D

The equations must be satisfied simultaneously, it is in general difficult to produce a diffracted beam with a fixed wavelength and a fixed crystal.

In 2D and 3D:

Lattice Planes

Bragg’s Law

sin2dBCAB

Bragg’s Law

If the path AB + CD is a multiple of the x-ray wavelength λ, then two waves will give a constructive interference:

nd sin2

sin2dCDABn

The diffracted waves will interfere destructively if equation is not satisfied.

Equation is called the Bragg equation and the angle θ is the Bragg angle.

n

Bragg’s Law

The incident beam and diffracted beam are always coplanar.

The angle between the diffracted beam and the transmitted beam is always 2.

Rewrite Bragg’s law:

1sin2

d

nSince sin 1: For n = 1: d2

UV radiation 500 Å

Cu K1 = 1.5406 ÅFor most crystals d ~ 3 Å 6 Å

aa

d

aa

d

a

lkh

d

2

1

002

001

1

2200

2100

2

222

2

sin2sin2 dn

d sin2d

Reciprocal lattice and Diffraction

0ss

0s

s

hkl

hkld

ss 1sin2 *0

d

321

*0 bbbdss

lkhhkl

It is equivalent to the Bragg law since

sin20 ss

sin2 hkld

*

hkld

Reciprocal lattice and Diffraction

hkl

hkld

ss 1sin2 *0

d

Sphere of Reflection – Ewald Sphere

sin22

1

2

1sin

1 *

hkl

hkl

hkl dd

dOC

*

hkldOB

Sphere of Reflection – Ewald Sphere

nd sin2

Silicon lattice constant:aSi = 5.43 Å

X-ray wavelength: = 1.5406 Å 2

222

2

1

a

lkh

d

For cubic crystal:Bragg’s law:

Kinematical x-ray diffraction

Two perovskites: SrTiO3 and CaTiO3

Differences: Peak position – d-spacing.

Peak intensity – atom type: Ca vs Sr.

aSTO = 3.905 Å

aCTO = 3.795 Å

O Ti

Sr/Ca

Scattering by an Electron

Elementary scattering unit in an atom is electron

Classical scattering by a single free electron – Thomson scattering equation:

If R = few cm:

2

2cos1 2

242

4

0

Rcm

eII

The polarization factor of an unpolarized primary beam

2

2cm26

42

4

1094.7 cm

e

26

0

10I

I

1 mg of matter has ~1020 electrons

Another way for electron to scatter is manifested in Compton effect.

Cullity p.127

Scattering by an Atom

electrononebyscatteredwavetheofamplitude

atomanbyscatteredwavetheofamplitudef

Atomic Scattering Factor

Scattering by an Atom

Scattering by a group of electrons at positions rn:

Scattering factor per electron:

dVife rss 0/2exp

Assuming spherical symmetry for the charge distribution = (r ) and taking origin at the center of the atom:

0

2 sin4 dr

kr

krrrfe

n

n

n

en drkr

krrrff

0

2 sin4

f – atomic scattering factor

Calling Z the number of electrons per atom we get:

Zdrrrn

n

0

24

sinf

For an atom containing several electrons:

sin4k

Scattering by an Atom

electrononebyscatteredwavetheofamplitude

atomanbyscatteredwavetheofamplitudef

The atomic scattering factor f = Z for any atom in the forward direction (2 = 0): I(2=0) =Z2

As increases f decreases functional dependence of the decrease depends on the details of the distribution of electrons around an atom (sometimes called the form factor)

f is calculated using quantum mechanics

Electron vs nuclear density

Powder diffraction patterns collected using Mo K radiation and neutron diffraction

Scattering by an Atom

Scattering by a Unit Cell

electronsinglebyscatteredamplitude

cellunit in atomsbyscatteredamplitudehklF

h

aACd

dMCN

h

h

00

0012 sin2

a

hx

ha

x

AC

ABMCN

AC

ABRBS

22

2

/

1313

13

For atoms A & C

For atoms A & B

hua

hx

2

213

If atom B position: axu /

For 3D: lwkvhu 2

phase

Scattering by a Unit Cell

We can write: lwkvhuii feAe 2

Nlwkvhui

n

Ni

n

iii

nnnn efefF

efefefF

1

2

1

321 ...321

2

hklhklhkl FFFI

Scattering by a Unit Cell

Examples

Unit cell has one atom at the origin

ffeF i 02

In this case the structure factor is independent of h, k and l ; it will decrease with fas sin/ increases (higher-order reflections)

Scattering by a Unit Cell

Examples

Unit cell is base-centered

khikhii effefeF 12/2/202

0

2

F

fF h and k unmixed

h and k mixed

(200), (400), (220)

(100), (121), (300)

224 fFhkl

02hklF “forbidden”

reflections

Body-Centered Unit Cell

Examples

For body-centered cell

lkhilkhilkhi effefeF 12/2/2/20002

0

2

F

fF when (h + k + l ) is even

when (h + k + l ) is odd

(200), (400), (220)

(100), (111), (300)

224 fFhkl

02hklF “forbidden”

reflections

Body-Centered Unit Cell

Examples

For body centered cell with different atoms:

lkhi

CsCl

lkhi

Cs

lkhi

Cl

eff

efefF

2/2/2/20002

CsCl

CsCl

ffF

ffF

when (h + k + l) is even

when (h + k + l) is odd

(200), (400), (220)

(100), (111), (300)

2ClCs

2ffFhkl

2ClCs

2ffFhkl

Cs+ Cl+

Face Centered Unit cell

The fcc crystal structure has atoms at 000, ½½0, ½0½ and 0½½:

lkilhikhiN

lwkvhui

nhkl eeefefF nnn

11

2

If h, k and l are all even or all odd numbers (“unmixed”), then the exponential terms all equal to +1 F = 4f

If h, k and l are mixed even and odd, then two of the exponential terms will equal -1 while one will equal +1 F = 0

2

hklF 16f 2, h, k and l unmixed even and odd

0, h, k and l mixed even and odd

The Structure Factor

The structure factor contains the information regarding the types (f ) and locations (u, v, w ) of atoms within a unit cell

A comparison of the observed and calculated structure factors is a common goal of X-ray structural analyses

The observed intensities must be corrected for experimental and geometric effects before these analyses can be performed

Nlwkvhui

nhklnnnefF

1

2

electronsingleabyscatteredamplitude

cellunitainatomsallbyscatteredamplitudehklF

Integrated Intensity

Structural factors: determined by crystal structure

Specimen factors: shape, size, grain size and distribution, microstructure

Instrumental factors: radiation properties, focusing geometry, type of detector

Peak intensity depends on

We can say that: 𝐼 𝑞 ∝ 𝐹 𝑞 2

Integrated Intensity

K – scale factor, required to normalize calculated and measured intensities.

phkl – multiplicity factor. Accounts for the presence of symmetrically equivalent points in reciprocal lattice.

L – Lorentz multiplier, defined by diffraction geometry.

P – polarization factor. Account for partial polarization of electromagnetic wave.

A – absorption multiplier. Accounts for incident and diffracted beam absorption.

Thkl – preferred orientation factor. Accounts for deviation from complete random grain distribution.

Ehkl – extinction multiplier. Accounts for deviation from kinematical diffraction model.

Fhkl – the structure factor. Defined by crystal structure of the material

2)()( qFETAPLpKqI hklhklhkl

The Multiplicity Factor

The multiplicity factor arises from the fact that in general there will be several sets of hkl -planes having different orientations in a crystal but with the same d and F 2

values

Evaluated by finding the number of variations in position and sign in h, k and land have planes with the same d and F 2

The value depends on hkl and crystal symmetry

For the highest cubic symmetry we have:

111,111,111,111,111,111,111,111

110,101,110,011,011,101,110,101,011,011,101,110

100,001,010,010,001,100 p100 = 6

p110 = 12

p111=8

The Polarization Factor

The polarization factor p arises from the fact that an electron does not scatter along its direction of vibration

In other directions electrons radiate with an intensity proportional to (sin )2:

The polarization factor (assuming that the incident beam is unpolarized):

2

2cos1 2 P

The Lorentz-Polarization Factor

The Lorenz factor L depends on the measurement technique used and, for the diffractometer data obtained by the usual θ-2θ or ω-2θ scans, it can be written as

The combination of geometric corrections are lumped together into a single Lorentz-polarization (LP ) factor:

The effect of the LP factor is to decrease the intensity at intermediate angles and increase the intensity in the forward and backwards directions

2sincos

1

2sinsin

1L

2

2

sincos

2cos1LP

The Absorption Factor

Angle-dependent absorption within the sample itself will modify the observed intensity

• Absorption factor for thin specimens is given by:

sin

2exp1

tA

where μ is the absorption coefficient, t is the total thickness of the film

• Absorption factor for infinite thickness specimen is:

2

A

The Extinction Factor

primary extinction secondary extinction

Extinction lowers the observed intensity of very strong reflections from perfect crystals

In powder diffraction usually this factor is smaller than experimental errors and therefore neglected

The Temperature Factor

As atoms vibrate about their equilibrium positions in a crystal, the electron density is spread out over a larger volume

This causes the atomic scattering factor to decrease with sin/ (or |S| = 4sin/ )more rapidly than it would normally:

2

2sin

B

e

where the thermal factor B is related to the mean square displacement of the atomic vibration:

228 uB

MM efeff 22

0 ~

The temperature factor is given by:

This is incorporated into the atomic scattering factor:

Sca

ttering b

y C

ato

m e

xpre

ssed in e

lect

rons

Diffracted Beam Intensity

2

hklhklhkl FFFI

bC IqFLPKApqI 2

)()()(

where K is the scaling factor, Ib is the background intensity, q = 4sinθ/λ is the

scattering vector for x-rays of wavelength λ

bC IqFt

KqI

2

2

2

)(sincos

2cos1

sin

2exp1)(

For thin films: